An immigration policy rewrite may be President Barack Obama’s top priority, but Senate Republicans are warning that if he tries to influence Congress too much, the delicate talks could run aground.

“I think this is going to be a congressional thing,” Senate Judiciary ranking member Charles E. Grassley, R-Iowa, said Wednesday. “I think the president is going to stay out of this. He doesn’t want to talk to Congress. You saw that last fall in the fiscal cliff.

Sen. Orrin G. Hatch, R-Utah, a veteran of previous immigration policy change efforts, said he hopes the president will use a light touch when it comes to pressing for his stated prerogatives.

“I actually believe he doesn’t care much for Congress,” Hatch said. The Utah lawmaker stressed that he likes “the president personally,” but he said Obama hasn’t reached out to lawmakers on recent legislative business such as the fiscal cliff.

“I hope we provide the leadership and that he follows along,” Hatch said.

So far on immigration, however, the White House has been willing to give Congress some leash on the issue, but only so much. Obama has already bristled at a bipartisan group of eight senators’ central compromise linking enforcement of border security improvements to any pathway to citizenship for undocumented immigrants.

The president is also pushing for protections for gay and lesbian couples over the objections of many Republicans, including Sen. John McCain, who is a key member of the group of eight senators who outlined a bipartisan immigration agreement Monday.

McCain warned Wednesday that including those kind of social issues could kill the bill.

“I think it is a red herring,” McCain said at a Politico Playbook breakfast. “I’m telling you if you load this up with social issues and things that are controversial then it will endanger the [bill].”

McCain continued, “I’ll be glad to talk about it ... but if someone views that as the most important aspect of comprehensive immigration reform, then we just have a fundamental disagreement.”

Charles E. Schumer, D-N.Y., one of the eight senators who appeared with McCain, said the issue of gay and lesbian couples will be one of many “on the table,” but he declined to speculate whether such protections would be in any final bill.

White House Press Secretary Jay Carney walked the line between openness to congressional tinkering and the president’s priorities Wednesday, but he was careful not to explicitly rain on anyone’s parade.

“There has to be a clear path, a path that ends in citizenship,” Carney said. But as for the insistence of Sen. Marco Rubio, R-Fla., also a part of the Senate group, and others of a link between enforcement and citizenship opportunities, he held his fire.

“We’re not going to prejudge, you know, legislation that hasn’t been written yet,” Carney said. “I don’t want to rule out or rule in” something that is not specific.

He reiterated that the president is committed to enhancing border security and enforcement.

Obama is also willing to consider a guest-worker program demanded by GOP senators such as Jeff Flake of Arizona, who is also a member of the group of eight, Carney said.

And on the issue of protections for gay and lesbian couples, Carney said the president’s position should not come as a surprise.

“The president has long believed that Americans with same-sex partners from other countries should not be faced with the painful choice between staying with the person they love or staying in the country they love,” he said.

It’s not yet clear, however, whether that’s something the White House might be willing to trade away, if needed, to secure votes for the larger bill.

And any Republicans wanting the president to lower his profile on the issue are sure to be disappointed. The White House clearly feels that its successes of the past few years have come when the president has marshaled support from across the country, rather than playing an inside Washington, D.C., salon game.

Obama also believes that Republicans need the legislation to pass for obvious demographic reasons, given the wave of Latino voters who supported the president’s re-election last year. Administration officials think the president’s postelection push already helped spur the group of senators to come to an accord.

Schumer, who said he has spoken with the president several times “face to face,” stressed that Obama is allowing Congress to do its work.

“He is rallying the country to do reform, getting us all together, but at the same time giving us the space to get something done, and I’ve been very impressed with not only his desire to get it done, but his ability to work with us as part of a team, as leader of the team, which he is, but to get it done,” Schumer said.